The circle diagram method

This is the heart of the chapter: the method for calculating the derivative of a composition of functions, such as $\sin(x^2)$, $e^{\sqrt{x}}$, $\arctan(1/x)$. The method rests on three ideas:

1. Represent the composition with a circle diagram;
2. Calculate the derivatives of the individual “links” of the chain using geometric symbols as placeholders;
3. Multiply the partial derivatives and substitute the symbols with the real expressions.

Given a composite function, one “decomposes” it into a chain of simple operations. Each operation is an arrow between two circles: the starting circle contains the input, the arriving one the result. The geometric symbols $\square$, $\triangle$ (and if needed $\bigcirc$, $\diamondsuit$…) indicate the content of the current circle without writing its expression explicitly.

Example — Diagram for $\sin(x^2)$

Circle diagram of the composition $x \to x^2 \to \sin(x^2)$: above each arrow the function applied (with the geometric symbol), below its derivative with respect to the symbol.

How to read the diagram:

• The circles contain the real expressions: $x$, then $x^2$, then $\sin(x^2)$.
• Above each arrow: the function applied, written with the geometric symbol $\square$ or $\triangle$ in place of the argument. Here "$\square^2$" means “square $\square$” (where $\square = x$); "$\sin(\triangle)$" means “take the sine of $\triangle$” (where $\triangle = x^2$).
• Below each arrow (in red): the derivative of that function with respect to its geometric symbol. Here the derivative of $\square^2$ is $2\square$; the derivative of $\sin\triangle$ is $\cos\triangle$.

Result: the chain rule is the product of the “below-arrow” terms, then one substitutes the symbols with the real expressions: $D[\sin(x^2)] = \underbrace{2\square}_{\square=x} \cdot \underbrace{\cos\triangle}_{\triangle=x^2} = 2x\cdot\cos(x^2).$

Links

Topics: Derivatives
Concepts: Composite derivative · Chain rule · Differentiation rules
Skills: Differentiating · Using formulae